Linear differential equations in the complex domain: problems of analytic continuation. Transl. from the Japanese by Yasutaka Sibuya.

*(English)*Zbl 1145.34378
Translations of Mathematical Monographs 82. Providence, RI: American Mathematical Society (ISBN 0-8218-4535-7). xiv, 267 p. (1990).

For the reviewer, who has been working on linear differential equations in the complex domain for most of his mathematical life, there used to be one book which (at first thought) came to his mind as an introduction to as well as a source of references in his field: W. R. Wasow’s Asymptotic expansions for ordinary differential equations [Interscience, New York (1965; Zbl 0133.35301)]. Now there is a second one: the present author’s translation of his Japanese monograph. In fact, the reviewer still treasures a photocopy of one chapter of the handwritten manuscript (in Japanese!), given to him by the author a long time ago. While this manuscript is amazingly beautiful to look at, the translation into English is certainly easier to read – in particular, if one does not know any Japanese.

Even the English translation may not be too easily readable for a student, or a person more oriented towards applications of the theory to other areas, since from the start a fair amount of (at least basic) knowledge in algebraic and cohomological methods is required. But for those who are at least somewhat familiar with such methods the book is an excellent source of information on more classical as well as recent results on such questions as the problem of standard forms (in the sense of G. D. Birkhoff), Riemann’s problem, the Stokes phenomenon, etc. Several appendices have been added to the Japanese version of the book in order to cover the most recent developments (Gevrey asymptotics, \(k\)-summability and isomonodromic deformations). The fact that even these appendices do not include anything on multi-summability (which was developed shortly after the book went into print) may serve as an indication of the rapid development presently going on in this area.

Altogether, the book can be highly recommended to everyone seriously interested in the area. It complements the more classical treatment in Wasow’s book by presenting results in a modern language, and is even more important as a source of information on very recent results.

In the preface to the Japanese edition the author leaves it to the reader to determine what grade Professor Kimura will give to the book. The reviewer would give it a straight A!

Even the English translation may not be too easily readable for a student, or a person more oriented towards applications of the theory to other areas, since from the start a fair amount of (at least basic) knowledge in algebraic and cohomological methods is required. But for those who are at least somewhat familiar with such methods the book is an excellent source of information on more classical as well as recent results on such questions as the problem of standard forms (in the sense of G. D. Birkhoff), Riemann’s problem, the Stokes phenomenon, etc. Several appendices have been added to the Japanese version of the book in order to cover the most recent developments (Gevrey asymptotics, \(k\)-summability and isomonodromic deformations). The fact that even these appendices do not include anything on multi-summability (which was developed shortly after the book went into print) may serve as an indication of the rapid development presently going on in this area.

Altogether, the book can be highly recommended to everyone seriously interested in the area. It complements the more classical treatment in Wasow’s book by presenting results in a modern language, and is even more important as a source of information on very recent results.

In the preface to the Japanese edition the author leaves it to the reader to determine what grade Professor Kimura will give to the book. The reviewer would give it a straight A!

Reviewer: W. Balser (MR1084379)